## Abstract

A discrete distribution D over Σ_{1} ×··· ×Σ_{n} is called (non-uniform) k -wise independent if for any subset of k indices {i_{1},...,i_{k}} and for any z_{1}∈Σi1,...,z_{k}∈Σik, Pr_{X~D}[Xi1···Xik = z_{1}···z_{k}] = Pr_{X~D}[Xi1 = z_{1}]···Pr_{X~D}[Xik = z_{k}]. We study the problem of testing (non-uniform) k -wise independent distributions over product spaces. For the uniform case we show an upper bound on the distance between a distribution D from k -wise independent distributions in terms of the sum of Fourier coefficients of D at vectors of weight at most k. Such a bound was previously known only when the underlying domain is {0,1}^{n}. For the non-uniform case, we give a new characterization of distributions being k -wise independent and further show that such a characterization is robust based on our results for the uniform case. These results greatly generalize those of Alon et al. (STOC'07, pp. 496-505) on uniform k -wise independence over the Boolean cubes to non-uniform k -wise independence over product spaces. Our results yield natural testing algorithms for k -wise independence with time and sample complexity sublinear in terms of the support size of the distribution when k is a constant. The main technical tools employed include discrete Fourier transform and the theory of linear systems of congruences.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013

Original language | English |
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Pages (from-to) | 265-312 |

Number of pages | 48 |

Journal | Random Structures and Algorithms |

Volume | 43 |

Issue number | 3 |

DOIs | |

State | Published - Oct 2013 |

## Keywords

- Discrete distributions
- Fourier analysis
- Property testing
- k -wise independence